It is crusty roll fit for my crusty role.

**Update: In case Fred show up, To use ratio and proportion correctly in this problem the 33C greenhouse effect would need to be equal to Surface radiation minus TOA radiation divide by four. 155/4=38.7 so either relationship in the equation is wrong, 33 is wrong or 155 is wrong, my money is on the 155.**

Just in case you are wondering, the emissivity estimate of 0.926 only accounts for part of the difference, 143.5wm-2, there is still nearly 10 Wm-2 missing.

Just in case you are wondering, the emissivity estimate of 0.926 only accounts for part of the difference, 143.5wm-2, there is still nearly 10 Wm-2 missing.

In the Monckton-Lucia online debate I am a non-combatant, I really don't care. I was curious if there was any justification for the linear approximation of flux versus temperature. Not only is there justification, there are several ways of using that approximation, with some pitfalls for the unwary. I will leave those issues to others. The one equation used was dF/dT=4F/T, so if I am going to use simple ratio and proportion to calculate a change in one relative to the other, it is a piece of cake. But is it valid with a mix of heat fluxes, latent thermal(sensible I prefer) and radiant?

I marked up this NASA energy budget drawing to determine the fluxes at the surface and atmosphere. There are a few differences between this drawing and the Kiehl and Trenberth energy budget drawings, mainly subtle differences.

If energy is fungible and the Earth's surface tends toward energy equilibrium, the sum of the energy fluxes at the surface will sum to the black body energy flux at 288K, the average surface temperature. Since the Earth has a greenhouse gas atmosphere, the temperature at the surface is greater than at the Top Of the Atmosphere (TOA). The difference in the heat flux at the surface is greater than the heat flux at the TOA by approximate 155W/m-2 on average. Adding twice the amount of CO2 to the atmosphere with increase the resistance to Outgoing Long wave Radiation (OLR)flow rate, or flux, changing the flux by 3.7Wm-2, which I am assuming is correct. As Co2 is a well mixed gas, the resistance will be greatest at the surface and decrease with altitude. Due to competition with water vapor in the lower atmosphere, the impact of CO2 will be more conductive and convective than radiative near the surface due to rapid transfer of absorbed energy by conduction (collisional transfer).

At the surface, all three basic types of heat flow, conduction(sensible), convection(a combination of sensible and latent) and radiative are present. To avoid confusion, conduction is used on the NASA cartoon while thermals is used on the K&T cartoons. Of the three types of heat flow, only radiative is directly changed by the addition of CO2. Of the radiative flux, only some spectral lines are effected by CO2, those spectral lines will cause the unaffected spectral lines to adjust to maintain equilibrium. Since energy is fungible, meaning it can change from one form to another as long as it is conserved, the conductive and latent fluxes with adjust to the change in radiative flux, limited by the thermodynamic relationships that influence them.

Based on the above the relationship Fout=Fc(onductive)+Fl(antent)+Fr(adiative) is true at the surface. Since only a portion of Fr(radiative) is effected, Fr can be separated into Fra(atmosphere) and Frs (space,) where Frs is the portion of the radiative spectrum not effected by the change in CO2.

For the modified drawing that yields, 390=24+79+(218+72). Fr=72+218=288 is a total of the heat loss from the surface interacting with the atmosphere only 21Wm-2 passes directly from the surface to space. The 390=24+79+21+267 would be the result of splitting Fr in Frs and Fra.

In order to use a ratio and proportion we have to define the range impacted by the change. The resistance to radiative flux in steady state creates a temperature imbalance between the surface (Ts) and the Top of the Amosphere (Toa) of Ts-Ttoa = Ts-255K or 33 degrees K, felt at the surface for an average temperature of 288K, the temperature and flux at the TOA will remain the same as it is in equilibrium with solar energy into the TOA. Then Tge=Ts-Ttoa. This range is equivalent to the change in radiative flux based on the black body temperature of the surface, 390-242=148 though 155W/m-2 is typically used after allowing for emissivity, which is the flux absorbed and re-emitted by the atmosphere, Fge.

So for a change in flux, (dFge/Fge) is proportional to d(Tge)/Tge, this can be written as dFge/dTge=4Fge/Tge => dTge=Tge*(dFge/4). For a discussion on this see, Uncertainty Monster Visits MIT. (This is a blog, so I am referencing a blog :)

Tge=33K, Fge=155W/m-2, dFge=2xCO2=3.7 and dTge is unknown

dTge=Tge*dFge/4=(33K*3.7)/(155Wm-2*4)=0.196K/Wm-2, based on the values typically assumed on the NASA drawing. For a 5 Wm-2 increase would produce 5Wm-2*0.196K/wm-2=0.98K, which is slightly lower than many estimates.

For the K&T old drawing, Fge=24+78+26=128Wm-2, Tge=33K and dFge=3.7 => 33*3.7/(155*4)=0.197 K/Wm-2

But this not the end of the story, Fr, the radiation from the surface, has the Frs portion that has a free pass to space and Fl and Fc have other thermodynamic issues to deal with. This will reduce the impact of Fge on Tge. In order to account for that impact we need to use Fnet = Fc+Fl+Fge+Fs

Fnet is complicated between the two drawings. In the NASA drawing, more radiation is shown leaving the surface and entering the atmosphere and less shown having a free path to space.

For NASA, Fnet = 51+21=72 and K&T 26 is calculated by converting the OLR to net and Fs is shown as 40 producing a Fnet of 66. So we have;

Nasa Fnet=24+79+51+21= 175 K&T Fnet=24+78+26+40= 168 Assuming Tge=33 and Fge=155 for both,

Nasa (33*3.7)/(4*175)=0.174Kwm-2 K&T (33*3.7)/(4*168)=0.18KWm-2

For a 5Wm-2 increase 5*0.174=0.87K

Both Fge and Fnet calculations should be considered as the increase in atmospheric temperature by Fge will feed back and the impact of that feedback will be reduced somewhat by the radiation window considered in Fnet. The truer value should be in between those two. Of the two drawings, the NASA is more realistic since water in liquid and solid formed in the atmosphere will emit in the atmospheric window to some degree.

Finally, by assuming that the net radiation is Fout-Fc-Fl=288 we can calculate,

(33*3.7)/(4*288)=0.106K/Wm-2 or we could use the total OLR based solely on the 228K radiation value of 390Wm-2 and obtain (33*3.7)/(4*390)=0.078K/Wm-2 In both cases there is a large amount of energy not actively participating either to warm the atmosphere or cool the surface. This produces an unrealistically low value for the Planck response. The atmospheric or greenhouse effect would respond only to an attempt to change the rate of flux through the atmosphere. The difference in temperature of the surface is due the effective temperature differences between the surface and the atmosphere which is in equilibrium if we assume the surface energy is seeking equilibrium, atmospheric energy is seeking equilibrium and the TOA is seeking equilibrium.

The assumption of equilibrium in no way implies true equilibrium. The impact of addition forcing on various layers of the atmosphere will have differing effects. An integration of the impact of different layers would have on their neighbors would be required for complete understanding. This method only provides a reasonable estimate of the Planck response to a change in emissivity. The varying limits of conductive, convective and latent responses would have to be considered separately.

While this has been interesting, a simple ratio and proportion of the surface is not all that valuable. It would better if used as part of an analysis of surface regions with similar feedback responses, like a tropical zone (slightly lower Planck response), northern hemisphere zone (higher Planck response with more water vapor feedback) and southern hemisphere zone (in between, but closer to the NH Planck response with less water vapor feedback). Each would have very different responses to a change in forcing. An atmospheric balance of each zone should also be considered as heat energy would migrate from one zone into another.

That's the end of me trying to act serious. Now its speculation time!

For these drawings, an atmospheric ratio is interesting. For the K$T old drawing, We have solar absorbed by clouds Fs=67, Fc(the thermals)=24, Fl=78 and Fr=26 for an Fnet=195. We do not have a Tge or Fge and temperatures in the atmosphere are well below the black body temperature at the TOA. We can calculate a change in forcing, 3.7/195=0.0036 and we have an estimate of the Planck response of 0.18K/Wm-2. Dividing the Planck response by the ratio of the change in forcing we get 50K. 288K-50K=238K, which is very close to the TOA effective temperature. If we use that as an estimate for Tge in the atmosphere, 50*3.7/(4*195)=0.23 or 5W/m-2*0.23=1.18K With an estimate of Tge for the atmosphere of 38K we would have the same Planck response as the surface. Subtracting that from the 288K surface temperature we would get an effective temperature of the atmosphere of -23 degrees C. This is roughly the potential temperature of a parcel of air at 250K and 600mb or an altitude of 4.5 km. Interesting?

UPDATE: The discrepancy between Tge and Fge is likely due to the effective emissivity plus a few assumptions. Fge should be approximately 133Wm-2 instead of the 155Wm-2 assumed, of course 288K at the surface is also assumed as well as 3.7Wm-2 being the change in forcing due to a doubling of CO2, so there is a whole lot of 'summin' goin' on. However, 22Wm-2 is a fair amount of missing heat for a high class radiation budget.

Now time to eat baked bacon bread.

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